Quadratic Functions and Equations

A quadratic function is defined by the general form f(x) = ax^2 + bx + c, where 'a', 'b', and 'c' are real constants and 'a' is not equal to zero. If 'a' were zero, the function would become linear. The graph of a quadratic function is a parabola, which is a symmetrical U-shaped curve. If 'a' > 0, the parabola opens upwards (has a minimum point), and if 'a' < 0, it opens downwards (has a maximum point).

A quadratic equation is formed when a quadratic function is set equal to zero: ax^2 + bx + c = 0. The solutions to this equation are called the roots (or zeros) of the quadratic. These roots correspond to the x-intercepts of the parabola, where the graph crosses the x-axis. Understanding the relationship between the function, its graph, and its equation is fundamental.

Factorisation is often the simplest method for solving quadratic equations, provided the quadratic expression can be factored. This method relies on the Null Factor Law, which states that if the product of two or more factors is zero, then at least one of the factors must be zero.

Steps for Factorisation:

1. Ensure the equation is in the form ax^2 + bx + c = 0.
2. Factorise the quadratic expression into two linear factors, e.g., (px + q)(rx + s) = 0.
3. Set each factor equal to zero and solve for x.

Example: Solve x^2 - 5x + 6 = 0.

• We look for two numbers that multiply to 6 and add to -5. These are -2 and -3.
• So, (x - 2)(x - 3) = 0.
• Setting each factor to zero: x - 2 = 0 => x = 2; x - 3 = 0 => x = 3.
• The roots are x = 2 and x = 3.

This method is efficient but not always applicable if the quadratic does not have simple integer factors.

Completing the square is a powerful method that can solve any quadratic equation and is also used to transform quadratic functions into vertex form. The goal is to rewrite the quadratic expression ax^2 + bx + c in the form a(x + p)^2 + q.

Steps for Completing the Square (for ax^2 + bx + c = 0):

1. Divide the entire equation by 'a' if a is not 1: x^2 + (b/a)x + (c/a) = 0.
2. Move the constant term to the right side: x^2 + (b/a)x = -c/a.
3. Take half of the coefficient of x ((b/a)/2 = b/2a), square it ((b/2a)^2), and add it to both sides of the equation.
   • x^2 + (b/a)x + (b/2a)^2 = -c/a + (b/2a)^2
4. Factor the left side as a perfect square: (x + b/2a)^2 = -c/a + (b/2a)^2.
5. Take the square root of both sides (remembering +/-) and solve for x.

Example: Solve x^2 + 6x + 5 = 0.

• x^2 + 6x = -5
• x^2 + 6x + (6/2)^2 = -5 + (6/2)^2
• x^2 + 6x + 9 = -5 + 9
• (x + 3)^2 = 4
• x + 3 = +/- sqrt(4)
• x + 3 = +/- 2
• x = -3 +/- 2, so x = -1 or x = -5.

This method is essential for understanding the derivation of the quadratic formula and for finding the vertex of a parabola.